Mr Musk said on his Twitter account that he was “up for a cage fight” with Mr Zuckerberg.
Mr Zuckerberg, the CEO of Meta, the parent company of Facebook and Instagram, subsequently tweeted a copy of Mr Musk’s message with the comment “send me location.”
“The story speaks for itself,” said a Meta representative to the BBC.
Mr Musk immediately responded to Mr Zuckerberg’s remark with the phrase “Vegas Octagon.”
The Octagon is the fenced-in enclosure and competition mat used for Ultimate Fighting Championship (UFC) fights. The UFC is headquartered in Las Vegas, Nevada.
Mr Musk, who will be 52 later this month, also tweeted: “I have this great move I call ‘The Walrus,’ where I just lie on top of my opponent and do nothing.”
Later, he posted short films of walruses, implying that his challenge to Mr Zuckerberg was not fully serious.
“I almost never work out, except for picking up my kids and throwing them in the air,” he said on Twitter.
Meanwhile, Mr Zuckerberg, 39, has started practising mixed martial arts (MMA) and has lately won jiu-jitsu contests.
When asked by the BBC for response, Twitter did not issue a remark.
The arguments have gone viral, with social media users arguing who would win the battle and others posting parodies such as made up ads advertising the fight.
For example, business consultant Seyi Taylor tweeted: “Choose your fighter” alongside images of the two tech titans.
However, asked who he thought would come out on top, he replied: “Zuckerberg all day! He’s 12 years younger. He is a lot smaller. I think he’s 5ft 7, Elon’s probably around 6ft. And Elon’s probably got a couple of stone in weight on him.
“But unfortunately Mr Musk has got no training whatsoever. Even though Zuckerberg’s only been training Brazilian jiu-jitsu for 18 months, it wouldn’t be difficult for him to take his back, wrap his arms around his neck and give him a good old cuddle and choke him out!”
